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Theorem xkoinjcn 21490

Description: Continuity of "injection", i.e. currying, as a function on continuous function spaces. (Contributed by Mario Carneiro, 23-Mar-2015.)

**Hypothesis**
Ref	Expression
xkoinjcn.3	⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩))

**Assertion**
Ref	Expression
xkoinjcn	⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → 𝐹 ∈ (𝑅 Cn ((𝑆 ×_t 𝑅) ^_ko 𝑆)))

Distinct variable groups: 𝑥,𝑦,𝑅 𝑥,𝑆,𝑦 𝑥,𝑌,𝑦 𝑥,𝑋,𝑦 Allowed substitution hints: 𝐹(𝑥,𝑦)

Proof of Theorem xkoinjcn Dummy variables 𝑓 𝑘 𝑟 𝑣 𝑤 𝑧 𝑡 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simplr 792	. . . 4 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑥 ∈ 𝑋) → 𝑆 ∈ (TopOn‘𝑌))
2	1	cnmptid 21464	. . . 4 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑥 ∈ 𝑋) → (𝑦 ∈ 𝑌 ↦ 𝑦) ∈ (𝑆 Cn 𝑆))
3		simpll 790	. . . . 5 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑥 ∈ 𝑋) → 𝑅 ∈ (TopOn‘𝑋))
4		simpr 477	. . . . 5 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
5	1, 3, 4	cnmptc 21465	. . . 4 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑥 ∈ 𝑋) → (𝑦 ∈ 𝑌 ↦ 𝑥) ∈ (𝑆 Cn 𝑅))
6	1, 2, 5	cnmpt1t 21468	. . 3 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ 𝑥 ∈ 𝑋) → (𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩) ∈ (𝑆 Cn (𝑆 ×_t 𝑅)))
7		xkoinjcn.3	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ (𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩))
8	6, 7	fmptd 6385	. 2 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → 𝐹:𝑋⟶(𝑆 Cn (𝑆 ×_t 𝑅)))
9		eqid 2622	. . . . . 6 ⊢ ∪ 𝑆 = ∪ 𝑆
10		eqid 2622	. . . . . 6 ⊢ {𝑤 ∈ 𝒫 ∪ 𝑆 ∣ (𝑆 ↾_t 𝑤) ∈ Comp} = {𝑤 ∈ 𝒫 ∪ 𝑆 ∣ (𝑆 ↾_t 𝑤) ∈ Comp}
11		eqid 2622	. . . . . 6 ⊢ (𝑘 ∈ {𝑤 ∈ 𝒫 ∪ 𝑆 ∣ (𝑆 ↾_t 𝑤) ∈ Comp}, 𝑣 ∈ (𝑆 ×_t 𝑅) ↦ {𝑓 ∈ (𝑆 Cn (𝑆 ×_t 𝑅)) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}) = (𝑘 ∈ {𝑤 ∈ 𝒫 ∪ 𝑆 ∣ (𝑆 ↾_t 𝑤) ∈ Comp}, 𝑣 ∈ (𝑆 ×_t 𝑅) ↦ {𝑓 ∈ (𝑆 Cn (𝑆 ×_t 𝑅)) ∣ (𝑓 “ 𝑘) ⊆ 𝑣})
12	9, 10, 11	xkobval 21389	. . . . 5 ⊢ ran (𝑘 ∈ {𝑤 ∈ 𝒫 ∪ 𝑆 ∣ (𝑆 ↾_t 𝑤) ∈ Comp}, 𝑣 ∈ (𝑆 ×_t 𝑅) ↦ {𝑓 ∈ (𝑆 Cn (𝑆 ×_t 𝑅)) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}) = {𝑧 ∣ ∃𝑘 ∈ 𝒫 ∪ 𝑆∃𝑣 ∈ (𝑆 ×_t 𝑅)((𝑆 ↾_t 𝑘) ∈ Comp ∧ 𝑧 = {𝑓 ∈ (𝑆 Cn (𝑆 ×_t 𝑅)) ∣ (𝑓 “ 𝑘) ⊆ 𝑣})}
13	12	abeq2i 2735	. . . 4 ⊢ (𝑧 ∈ ran (𝑘 ∈ {𝑤 ∈ 𝒫 ∪ 𝑆 ∣ (𝑆 ↾_t 𝑤) ∈ Comp}, 𝑣 ∈ (𝑆 ×_t 𝑅) ↦ {𝑓 ∈ (𝑆 Cn (𝑆 ×_t 𝑅)) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}) ↔ ∃𝑘 ∈ 𝒫 ∪ 𝑆∃𝑣 ∈ (𝑆 ×_t 𝑅)((𝑆 ↾_t 𝑘) ∈ Comp ∧ 𝑧 = {𝑓 ∈ (𝑆 Cn (𝑆 ×_t 𝑅)) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}))
14		simpll 790	. . . . . . . . . . . 12 ⊢ ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) → (𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)))
15	14, 6	sylan 488	. . . . . . . . . . 11 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ 𝑥 ∈ 𝑋) → (𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩) ∈ (𝑆 Cn (𝑆 ×_t 𝑅)))
16		imaeq1 5461	. . . . . . . . . . . . 13 ⊢ (𝑓 = (𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩) → (𝑓 “ 𝑘) = ((𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩) “ 𝑘))
17	16	sseq1d 3632	. . . . . . . . . . . 12 ⊢ (𝑓 = (𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩) → ((𝑓 “ 𝑘) ⊆ 𝑣 ↔ ((𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩) “ 𝑘) ⊆ 𝑣))
18	17	elrab3 3364	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩) ∈ (𝑆 Cn (𝑆 ×_t 𝑅)) → ((𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩) ∈ {𝑓 ∈ (𝑆 Cn (𝑆 ×_t 𝑅)) ∣ (𝑓 “ 𝑘) ⊆ 𝑣} ↔ ((𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩) “ 𝑘) ⊆ 𝑣))
19	15, 18	syl 17	. . . . . . . . . 10 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ 𝑥 ∈ 𝑋) → ((𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩) ∈ {𝑓 ∈ (𝑆 Cn (𝑆 ×_t 𝑅)) ∣ (𝑓 “ 𝑘) ⊆ 𝑣} ↔ ((𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩) “ 𝑘) ⊆ 𝑣))
20		funmpt 5926	. . . . . . . . . . 11 ⊢ Fun (𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩)
21		simplrl 800	. . . . . . . . . . . . . . 15 ⊢ ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) → 𝑘 ∈ 𝒫 ∪ 𝑆)
22	21	elpwid 4170	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) → 𝑘 ⊆ ∪ 𝑆)
23	14	simprd 479	. . . . . . . . . . . . . . 15 ⊢ ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) → 𝑆 ∈ (TopOn‘𝑌))
24		toponuni 20719	. . . . . . . . . . . . . . 15 ⊢ (𝑆 ∈ (TopOn‘𝑌) → 𝑌 = ∪ 𝑆)
25	23, 24	syl 17	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) → 𝑌 = ∪ 𝑆)
26	22, 25	sseqtr4d 3642	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) → 𝑘 ⊆ 𝑌)
27	26	adantr 481	. . . . . . . . . . . 12 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ 𝑥 ∈ 𝑋) → 𝑘 ⊆ 𝑌)
28		dmmptg 5632	. . . . . . . . . . . . 13 ⊢ (∀𝑦 ∈ 𝑌 ⟨𝑦, 𝑥⟩ ∈ V → dom (𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩) = 𝑌)
29		opex 4932	. . . . . . . . . . . . . 14 ⊢ ⟨𝑦, 𝑥⟩ ∈ V
30	29	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝑌 → ⟨𝑦, 𝑥⟩ ∈ V)
31	28, 30	mprg 2926	. . . . . . . . . . . 12 ⊢ dom (𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩) = 𝑌
32	27, 31	syl6sseqr 3652	. . . . . . . . . . 11 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ 𝑥 ∈ 𝑋) → 𝑘 ⊆ dom (𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩))
33		funimass4 6247	. . . . . . . . . . 11 ⊢ ((Fun (𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩) ∧ 𝑘 ⊆ dom (𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩)) → (((𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩) “ 𝑘) ⊆ 𝑣 ↔ ∀𝑧 ∈ 𝑘 ((𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩)‘𝑧) ∈ 𝑣))
34	20, 32, 33	sylancr 695	. . . . . . . . . 10 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ 𝑥 ∈ 𝑋) → (((𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩) “ 𝑘) ⊆ 𝑣 ↔ ∀𝑧 ∈ 𝑘 ((𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩)‘𝑧) ∈ 𝑣))
35	27	sselda 3603	. . . . . . . . . . . . . . 15 ⊢ ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ 𝑥 ∈ 𝑋) ∧ 𝑧 ∈ 𝑘) → 𝑧 ∈ 𝑌)
36		opeq1 4402	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑧 → ⟨𝑦, 𝑥⟩ = ⟨𝑧, 𝑥⟩)
37		eqid 2622	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩) = (𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩)
38		opex 4932	. . . . . . . . . . . . . . . 16 ⊢ ⟨𝑧, 𝑥⟩ ∈ V
39	36, 37, 38	fvmpt 6282	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ 𝑌 → ((𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩)‘𝑧) = ⟨𝑧, 𝑥⟩)
40	35, 39	syl 17	. . . . . . . . . . . . . 14 ⊢ ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ 𝑥 ∈ 𝑋) ∧ 𝑧 ∈ 𝑘) → ((𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩)‘𝑧) = ⟨𝑧, 𝑥⟩)
41	40	eleq1d 2686	. . . . . . . . . . . . 13 ⊢ ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ 𝑥 ∈ 𝑋) ∧ 𝑧 ∈ 𝑘) → (((𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩)‘𝑧) ∈ 𝑣 ↔ ⟨𝑧, 𝑥⟩ ∈ 𝑣))
42		vex 3203	. . . . . . . . . . . . . 14 ⊢ 𝑥 ∈ V
43		opeq2 4403	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = 𝑥 → ⟨𝑧, 𝑤⟩ = ⟨𝑧, 𝑥⟩)
44	43	eleq1d 2686	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑥 → (⟨𝑧, 𝑤⟩ ∈ 𝑣 ↔ ⟨𝑧, 𝑥⟩ ∈ 𝑣))
45	42, 44	ralsn 4222	. . . . . . . . . . . . 13 ⊢ (∀𝑤 ∈ {𝑥}⟨𝑧, 𝑤⟩ ∈ 𝑣 ↔ ⟨𝑧, 𝑥⟩ ∈ 𝑣)
46	41, 45	syl6bbr 278	. . . . . . . . . . . 12 ⊢ ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ 𝑥 ∈ 𝑋) ∧ 𝑧 ∈ 𝑘) → (((𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩)‘𝑧) ∈ 𝑣 ↔ ∀𝑤 ∈ {𝑥}⟨𝑧, 𝑤⟩ ∈ 𝑣))
47	46	ralbidva 2985	. . . . . . . . . . 11 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ 𝑥 ∈ 𝑋) → (∀𝑧 ∈ 𝑘 ((𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩)‘𝑧) ∈ 𝑣 ↔ ∀𝑧 ∈ 𝑘 ∀𝑤 ∈ {𝑥}⟨𝑧, 𝑤⟩ ∈ 𝑣))
48		dfss3 3592	. . . . . . . . . . . 12 ⊢ ((𝑘 × {𝑥}) ⊆ 𝑣 ↔ ∀𝑡 ∈ (𝑘 × {𝑥})𝑡 ∈ 𝑣)
49		eleq1 2689	. . . . . . . . . . . . 13 ⊢ (𝑡 = ⟨𝑧, 𝑤⟩ → (𝑡 ∈ 𝑣 ↔ ⟨𝑧, 𝑤⟩ ∈ 𝑣))
50	49	ralxp 5263	. . . . . . . . . . . 12 ⊢ (∀𝑡 ∈ (𝑘 × {𝑥})𝑡 ∈ 𝑣 ↔ ∀𝑧 ∈ 𝑘 ∀𝑤 ∈ {𝑥}⟨𝑧, 𝑤⟩ ∈ 𝑣)
51	48, 50	bitri 264	. . . . . . . . . . 11 ⊢ ((𝑘 × {𝑥}) ⊆ 𝑣 ↔ ∀𝑧 ∈ 𝑘 ∀𝑤 ∈ {𝑥}⟨𝑧, 𝑤⟩ ∈ 𝑣)
52	47, 51	syl6bbr 278	. . . . . . . . . 10 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ 𝑥 ∈ 𝑋) → (∀𝑧 ∈ 𝑘 ((𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩)‘𝑧) ∈ 𝑣 ↔ (𝑘 × {𝑥}) ⊆ 𝑣))
53	19, 34, 52	3bitrd 294	. . . . . . . . 9 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ 𝑥 ∈ 𝑋) → ((𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩) ∈ {𝑓 ∈ (𝑆 Cn (𝑆 ×_t 𝑅)) ∣ (𝑓 “ 𝑘) ⊆ 𝑣} ↔ (𝑘 × {𝑥}) ⊆ 𝑣))
54	53	rabbidva 3188	. . . . . . . 8 ⊢ ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) → {𝑥 ∈ 𝑋 ∣ (𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩) ∈ {𝑓 ∈ (𝑆 Cn (𝑆 ×_t 𝑅)) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}} = {𝑥 ∈ 𝑋 ∣ (𝑘 × {𝑥}) ⊆ 𝑣})
55		sneq 4187	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑤 → {𝑥} = {𝑤})
56	55	xpeq2d 5139	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑤 → (𝑘 × {𝑥}) = (𝑘 × {𝑤}))
57	56	sseq1d 3632	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑤 → ((𝑘 × {𝑥}) ⊆ 𝑣 ↔ (𝑘 × {𝑤}) ⊆ 𝑣))
58	57	elrab 3363	. . . . . . . . . . 11 ⊢ (𝑤 ∈ {𝑥 ∈ 𝑋 ∣ (𝑘 × {𝑥}) ⊆ 𝑣} ↔ (𝑤 ∈ 𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣))
59		eqid 2622	. . . . . . . . . . . . 13 ⊢ ∪ (𝑆 ↾_t 𝑘) = ∪ (𝑆 ↾_t 𝑘)
60		eqid 2622	. . . . . . . . . . . . 13 ⊢ ∪ 𝑅 = ∪ 𝑅
61		simplr 792	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ (𝑤 ∈ 𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) → (𝑆 ↾_t 𝑘) ∈ Comp)
62		simpll 790	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) → 𝑅 ∈ (TopOn‘𝑋))
63	62	ad2antrr 762	. . . . . . . . . . . . . 14 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ (𝑤 ∈ 𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) → 𝑅 ∈ (TopOn‘𝑋))
64		topontop 20718	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ (TopOn‘𝑋) → 𝑅 ∈ Top)
65	63, 64	syl 17	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ (𝑤 ∈ 𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) → 𝑅 ∈ Top)
66		topontop 20718	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑆 ∈ (TopOn‘𝑌) → 𝑆 ∈ Top)
67	66	adantl 482	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → 𝑆 ∈ Top)
68	64	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → 𝑅 ∈ Top)
69		txtop 21372	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑆 ∈ Top ∧ 𝑅 ∈ Top) → (𝑆 ×_t 𝑅) ∈ Top)
70	67, 68, 69	syl2anc 693	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑆 ×_t 𝑅) ∈ Top)
71	70	ad3antrrr 766	. . . . . . . . . . . . . . 15 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ (𝑤 ∈ 𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) → (𝑆 ×_t 𝑅) ∈ Top)
72		vex 3203	. . . . . . . . . . . . . . . 16 ⊢ 𝑘 ∈ V
73		toponmax 20730	. . . . . . . . . . . . . . . . 17 ⊢ (𝑅 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝑅)
74	63, 73	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ (𝑤 ∈ 𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) → 𝑋 ∈ 𝑅)
75		xpexg 6960	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ V ∧ 𝑋 ∈ 𝑅) → (𝑘 × 𝑋) ∈ V)
76	72, 74, 75	sylancr 695	. . . . . . . . . . . . . . 15 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ (𝑤 ∈ 𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) → (𝑘 × 𝑋) ∈ V)
77		simprr 796	. . . . . . . . . . . . . . . 16 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) → 𝑣 ∈ (𝑆 ×_t 𝑅))
78	77	ad2antrr 762	. . . . . . . . . . . . . . 15 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ (𝑤 ∈ 𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) → 𝑣 ∈ (𝑆 ×_t 𝑅))
79		elrestr 16089	. . . . . . . . . . . . . . 15 ⊢ (((𝑆 ×_t 𝑅) ∈ Top ∧ (𝑘 × 𝑋) ∈ V ∧ 𝑣 ∈ (𝑆 ×_t 𝑅)) → (𝑣 ∩ (𝑘 × 𝑋)) ∈ ((𝑆 ×_t 𝑅) ↾_t (𝑘 × 𝑋)))
80	71, 76, 78, 79	syl3anc 1326	. . . . . . . . . . . . . 14 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ (𝑤 ∈ 𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) → (𝑣 ∩ (𝑘 × 𝑋)) ∈ ((𝑆 ×_t 𝑅) ↾_t (𝑘 × 𝑋)))
81	67	ad3antrrr 766	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ (𝑤 ∈ 𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) → 𝑆 ∈ Top)
82	72	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ (𝑤 ∈ 𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) → 𝑘 ∈ V)
83		txrest 21434	. . . . . . . . . . . . . . . 16 ⊢ (((𝑆 ∈ Top ∧ 𝑅 ∈ Top) ∧ (𝑘 ∈ V ∧ 𝑋 ∈ 𝑅)) → ((𝑆 ×_t 𝑅) ↾_t (𝑘 × 𝑋)) = ((𝑆 ↾_t 𝑘) ×_t (𝑅 ↾_t 𝑋)))
84	81, 65, 82, 74, 83	syl22anc 1327	. . . . . . . . . . . . . . 15 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ (𝑤 ∈ 𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) → ((𝑆 ×_t 𝑅) ↾_t (𝑘 × 𝑋)) = ((𝑆 ↾_t 𝑘) ×_t (𝑅 ↾_t 𝑋)))
85		toponuni 20719	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑅 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝑅)
86	63, 85	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ (𝑤 ∈ 𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) → 𝑋 = ∪ 𝑅)
87	86	oveq2d 6666	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ (𝑤 ∈ 𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) → (𝑅 ↾_t 𝑋) = (𝑅 ↾_t ∪ 𝑅))
88	60	restid 16094	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑅 ∈ (TopOn‘𝑋) → (𝑅 ↾_t ∪ 𝑅) = 𝑅)
89	63, 88	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ (𝑤 ∈ 𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) → (𝑅 ↾_t ∪ 𝑅) = 𝑅)
90	87, 89	eqtrd 2656	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ (𝑤 ∈ 𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) → (𝑅 ↾_t 𝑋) = 𝑅)
91	90	oveq2d 6666	. . . . . . . . . . . . . . 15 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ (𝑤 ∈ 𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) → ((𝑆 ↾_t 𝑘) ×_t (𝑅 ↾_t 𝑋)) = ((𝑆 ↾_t 𝑘) ×_t 𝑅))
92	84, 91	eqtrd 2656	. . . . . . . . . . . . . 14 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ (𝑤 ∈ 𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) → ((𝑆 ×_t 𝑅) ↾_t (𝑘 × 𝑋)) = ((𝑆 ↾_t 𝑘) ×_t 𝑅))
93	80, 92	eleqtrd 2703	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ (𝑤 ∈ 𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) → (𝑣 ∩ (𝑘 × 𝑋)) ∈ ((𝑆 ↾_t 𝑘) ×_t 𝑅))
94	23	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ (𝑤 ∈ 𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) → 𝑆 ∈ (TopOn‘𝑌))
95	26	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ (𝑤 ∈ 𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) → 𝑘 ⊆ 𝑌)
96		resttopon 20965	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑆 ∈ (TopOn‘𝑌) ∧ 𝑘 ⊆ 𝑌) → (𝑆 ↾_t 𝑘) ∈ (TopOn‘𝑘))
97	94, 95, 96	syl2anc 693	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ (𝑤 ∈ 𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) → (𝑆 ↾_t 𝑘) ∈ (TopOn‘𝑘))
98		toponuni 20719	. . . . . . . . . . . . . . . 16 ⊢ ((𝑆 ↾_t 𝑘) ∈ (TopOn‘𝑘) → 𝑘 = ∪ (𝑆 ↾_t 𝑘))
99	97, 98	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ (𝑤 ∈ 𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) → 𝑘 = ∪ (𝑆 ↾_t 𝑘))
100	99	xpeq1d 5138	. . . . . . . . . . . . . 14 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ (𝑤 ∈ 𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) → (𝑘 × {𝑤}) = (∪ (𝑆 ↾_t 𝑘) × {𝑤}))
101		simprr 796	. . . . . . . . . . . . . . 15 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ (𝑤 ∈ 𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) → (𝑘 × {𝑤}) ⊆ 𝑣)
102		simprl 794	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ (𝑤 ∈ 𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) → 𝑤 ∈ 𝑋)
103	102	snssd 4340	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ (𝑤 ∈ 𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) → {𝑤} ⊆ 𝑋)
104		xpss2 5229	. . . . . . . . . . . . . . . 16 ⊢ ({𝑤} ⊆ 𝑋 → (𝑘 × {𝑤}) ⊆ (𝑘 × 𝑋))
105	103, 104	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ (𝑤 ∈ 𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) → (𝑘 × {𝑤}) ⊆ (𝑘 × 𝑋))
106	101, 105	ssind 3837	. . . . . . . . . . . . . 14 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ (𝑤 ∈ 𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) → (𝑘 × {𝑤}) ⊆ (𝑣 ∩ (𝑘 × 𝑋)))
107	100, 106	eqsstr3d 3640	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ (𝑤 ∈ 𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) → (∪ (𝑆 ↾_t 𝑘) × {𝑤}) ⊆ (𝑣 ∩ (𝑘 × 𝑋)))
108	102, 86	eleqtrd 2703	. . . . . . . . . . . . 13 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ (𝑤 ∈ 𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) → 𝑤 ∈ ∪ 𝑅)
109	59, 60, 61, 65, 93, 107, 108	txtube 21443	. . . . . . . . . . . 12 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ (𝑤 ∈ 𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) → ∃𝑟 ∈ 𝑅 (𝑤 ∈ 𝑟 ∧ (∪ (𝑆 ↾_t 𝑘) × 𝑟) ⊆ (𝑣 ∩ (𝑘 × 𝑋))))
110		toponss 20731	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑟 ∈ 𝑅) → 𝑟 ⊆ 𝑋)
111	63, 110	sylan 488	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ (𝑤 ∈ 𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) ∧ 𝑟 ∈ 𝑅) → 𝑟 ⊆ 𝑋)
112		ssrab 3680	. . . . . . . . . . . . . . . . 17 ⊢ (𝑟 ⊆ {𝑥 ∈ 𝑋 ∣ (𝑘 × {𝑥}) ⊆ 𝑣} ↔ (𝑟 ⊆ 𝑋 ∧ ∀𝑥 ∈ 𝑟 (𝑘 × {𝑥}) ⊆ 𝑣))
113	112	baib 944	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 ⊆ 𝑋 → (𝑟 ⊆ {𝑥 ∈ 𝑋 ∣ (𝑘 × {𝑥}) ⊆ 𝑣} ↔ ∀𝑥 ∈ 𝑟 (𝑘 × {𝑥}) ⊆ 𝑣))
114	111, 113	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ (𝑤 ∈ 𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) ∧ 𝑟 ∈ 𝑅) → (𝑟 ⊆ {𝑥 ∈ 𝑋 ∣ (𝑘 × {𝑥}) ⊆ 𝑣} ↔ ∀𝑥 ∈ 𝑟 (𝑘 × {𝑥}) ⊆ 𝑣))
115		xpss2 5229	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑟 ⊆ 𝑋 → (𝑘 × 𝑟) ⊆ (𝑘 × 𝑋))
116	111, 115	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ (𝑤 ∈ 𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) ∧ 𝑟 ∈ 𝑅) → (𝑘 × 𝑟) ⊆ (𝑘 × 𝑋))
117	116	biantrud 528	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ (𝑤 ∈ 𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) ∧ 𝑟 ∈ 𝑅) → ((𝑘 × 𝑟) ⊆ 𝑣 ↔ ((𝑘 × 𝑟) ⊆ 𝑣 ∧ (𝑘 × 𝑟) ⊆ (𝑘 × 𝑋))))
118		iunid 4575	. . . . . . . . . . . . . . . . . . . 20 ⊢ ∪ 𝑥 ∈ 𝑟 {𝑥} = 𝑟
119	118	xpeq2i 5136	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 × ∪ 𝑥 ∈ 𝑟 {𝑥}) = (𝑘 × 𝑟)
120		xpiundi 5173	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 × ∪ 𝑥 ∈ 𝑟 {𝑥}) = ∪ 𝑥 ∈ 𝑟 (𝑘 × {𝑥})
121	119, 120	eqtr3i 2646	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 × 𝑟) = ∪ 𝑥 ∈ 𝑟 (𝑘 × {𝑥})
122	121	sseq1i 3629	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑘 × 𝑟) ⊆ 𝑣 ↔ ∪ 𝑥 ∈ 𝑟 (𝑘 × {𝑥}) ⊆ 𝑣)
123		iunss 4561	. . . . . . . . . . . . . . . . 17 ⊢ (∪ 𝑥 ∈ 𝑟 (𝑘 × {𝑥}) ⊆ 𝑣 ↔ ∀𝑥 ∈ 𝑟 (𝑘 × {𝑥}) ⊆ 𝑣)
124	122, 123	bitri 264	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 × 𝑟) ⊆ 𝑣 ↔ ∀𝑥 ∈ 𝑟 (𝑘 × {𝑥}) ⊆ 𝑣)
125		ssin 3835	. . . . . . . . . . . . . . . 16 ⊢ (((𝑘 × 𝑟) ⊆ 𝑣 ∧ (𝑘 × 𝑟) ⊆ (𝑘 × 𝑋)) ↔ (𝑘 × 𝑟) ⊆ (𝑣 ∩ (𝑘 × 𝑋)))
126	117, 124, 125	3bitr3g 302	. . . . . . . . . . . . . . 15 ⊢ ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ (𝑤 ∈ 𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) ∧ 𝑟 ∈ 𝑅) → (∀𝑥 ∈ 𝑟 (𝑘 × {𝑥}) ⊆ 𝑣 ↔ (𝑘 × 𝑟) ⊆ (𝑣 ∩ (𝑘 × 𝑋))))
127	99	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ (𝑤 ∈ 𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) ∧ 𝑟 ∈ 𝑅) → 𝑘 = ∪ (𝑆 ↾_t 𝑘))
128	127	xpeq1d 5138	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ (𝑤 ∈ 𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) ∧ 𝑟 ∈ 𝑅) → (𝑘 × 𝑟) = (∪ (𝑆 ↾_t 𝑘) × 𝑟))
129	128	sseq1d 3632	. . . . . . . . . . . . . . 15 ⊢ ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ (𝑤 ∈ 𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) ∧ 𝑟 ∈ 𝑅) → ((𝑘 × 𝑟) ⊆ (𝑣 ∩ (𝑘 × 𝑋)) ↔ (∪ (𝑆 ↾_t 𝑘) × 𝑟) ⊆ (𝑣 ∩ (𝑘 × 𝑋))))
130	114, 126, 129	3bitrd 294	. . . . . . . . . . . . . 14 ⊢ ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ (𝑤 ∈ 𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) ∧ 𝑟 ∈ 𝑅) → (𝑟 ⊆ {𝑥 ∈ 𝑋 ∣ (𝑘 × {𝑥}) ⊆ 𝑣} ↔ (∪ (𝑆 ↾_t 𝑘) × 𝑟) ⊆ (𝑣 ∩ (𝑘 × 𝑋))))
131	130	anbi2d 740	. . . . . . . . . . . . 13 ⊢ ((((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ (𝑤 ∈ 𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) ∧ 𝑟 ∈ 𝑅) → ((𝑤 ∈ 𝑟 ∧ 𝑟 ⊆ {𝑥 ∈ 𝑋 ∣ (𝑘 × {𝑥}) ⊆ 𝑣}) ↔ (𝑤 ∈ 𝑟 ∧ (∪ (𝑆 ↾_t 𝑘) × 𝑟) ⊆ (𝑣 ∩ (𝑘 × 𝑋)))))
132	131	rexbidva 3049	. . . . . . . . . . . 12 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ (𝑤 ∈ 𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) → (∃𝑟 ∈ 𝑅 (𝑤 ∈ 𝑟 ∧ 𝑟 ⊆ {𝑥 ∈ 𝑋 ∣ (𝑘 × {𝑥}) ⊆ 𝑣}) ↔ ∃𝑟 ∈ 𝑅 (𝑤 ∈ 𝑟 ∧ (∪ (𝑆 ↾_t 𝑘) × 𝑟) ⊆ (𝑣 ∩ (𝑘 × 𝑋)))))
133	109, 132	mpbird 247	. . . . . . . . . . 11 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ (𝑤 ∈ 𝑋 ∧ (𝑘 × {𝑤}) ⊆ 𝑣)) → ∃𝑟 ∈ 𝑅 (𝑤 ∈ 𝑟 ∧ 𝑟 ⊆ {𝑥 ∈ 𝑋 ∣ (𝑘 × {𝑥}) ⊆ 𝑣}))
134	58, 133	sylan2b 492	. . . . . . . . . 10 ⊢ (((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) ∧ 𝑤 ∈ {𝑥 ∈ 𝑋 ∣ (𝑘 × {𝑥}) ⊆ 𝑣}) → ∃𝑟 ∈ 𝑅 (𝑤 ∈ 𝑟 ∧ 𝑟 ⊆ {𝑥 ∈ 𝑋 ∣ (𝑘 × {𝑥}) ⊆ 𝑣}))
135	134	ralrimiva 2966	. . . . . . . . 9 ⊢ ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) → ∀𝑤 ∈ {𝑥 ∈ 𝑋 ∣ (𝑘 × {𝑥}) ⊆ 𝑣}∃𝑟 ∈ 𝑅 (𝑤 ∈ 𝑟 ∧ 𝑟 ⊆ {𝑥 ∈ 𝑋 ∣ (𝑘 × {𝑥}) ⊆ 𝑣}))
136		eltop2 20779	. . . . . . . . . 10 ⊢ (𝑅 ∈ Top → ({𝑥 ∈ 𝑋 ∣ (𝑘 × {𝑥}) ⊆ 𝑣} ∈ 𝑅 ↔ ∀𝑤 ∈ {𝑥 ∈ 𝑋 ∣ (𝑘 × {𝑥}) ⊆ 𝑣}∃𝑟 ∈ 𝑅 (𝑤 ∈ 𝑟 ∧ 𝑟 ⊆ {𝑥 ∈ 𝑋 ∣ (𝑘 × {𝑥}) ⊆ 𝑣})))
137	14, 68, 136	3syl 18	. . . . . . . . 9 ⊢ ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) → ({𝑥 ∈ 𝑋 ∣ (𝑘 × {𝑥}) ⊆ 𝑣} ∈ 𝑅 ↔ ∀𝑤 ∈ {𝑥 ∈ 𝑋 ∣ (𝑘 × {𝑥}) ⊆ 𝑣}∃𝑟 ∈ 𝑅 (𝑤 ∈ 𝑟 ∧ 𝑟 ⊆ {𝑥 ∈ 𝑋 ∣ (𝑘 × {𝑥}) ⊆ 𝑣})))
138	135, 137	mpbird 247	. . . . . . . 8 ⊢ ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) → {𝑥 ∈ 𝑋 ∣ (𝑘 × {𝑥}) ⊆ 𝑣} ∈ 𝑅)
139	54, 138	eqeltrd 2701	. . . . . . 7 ⊢ ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) → {𝑥 ∈ 𝑋 ∣ (𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩) ∈ {𝑓 ∈ (𝑆 Cn (𝑆 ×_t 𝑅)) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}} ∈ 𝑅)
140		imaeq2 5462	. . . . . . . . 9 ⊢ (𝑧 = {𝑓 ∈ (𝑆 Cn (𝑆 ×_t 𝑅)) ∣ (𝑓 “ 𝑘) ⊆ 𝑣} → (^◡𝐹 “ 𝑧) = (^◡𝐹 “ {𝑓 ∈ (𝑆 Cn (𝑆 ×_t 𝑅)) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}))
141	7	mptpreima 5628	. . . . . . . . 9 ⊢ (^◡𝐹 “ {𝑓 ∈ (𝑆 Cn (𝑆 ×_t 𝑅)) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}) = {𝑥 ∈ 𝑋 ∣ (𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩) ∈ {𝑓 ∈ (𝑆 Cn (𝑆 ×_t 𝑅)) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}}
142	140, 141	syl6eq 2672	. . . . . . . 8 ⊢ (𝑧 = {𝑓 ∈ (𝑆 Cn (𝑆 ×_t 𝑅)) ∣ (𝑓 “ 𝑘) ⊆ 𝑣} → (^◡𝐹 “ 𝑧) = {𝑥 ∈ 𝑋 ∣ (𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩) ∈ {𝑓 ∈ (𝑆 Cn (𝑆 ×_t 𝑅)) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}})
143	142	eleq1d 2686	. . . . . . 7 ⊢ (𝑧 = {𝑓 ∈ (𝑆 Cn (𝑆 ×_t 𝑅)) ∣ (𝑓 “ 𝑘) ⊆ 𝑣} → ((^◡𝐹 “ 𝑧) ∈ 𝑅 ↔ {𝑥 ∈ 𝑋 ∣ (𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩) ∈ {𝑓 ∈ (𝑆 Cn (𝑆 ×_t 𝑅)) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}} ∈ 𝑅))
144	139, 143	syl5ibrcom 237	. . . . . 6 ⊢ ((((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) ∧ (𝑆 ↾_t 𝑘) ∈ Comp) → (𝑧 = {𝑓 ∈ (𝑆 Cn (𝑆 ×_t 𝑅)) ∣ (𝑓 “ 𝑘) ⊆ 𝑣} → (^◡𝐹 “ 𝑧) ∈ 𝑅))
145	144	expimpd 629	. . . . 5 ⊢ (((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) ∧ (𝑘 ∈ 𝒫 ∪ 𝑆 ∧ 𝑣 ∈ (𝑆 ×_t 𝑅))) → (((𝑆 ↾_t 𝑘) ∈ Comp ∧ 𝑧 = {𝑓 ∈ (𝑆 Cn (𝑆 ×_t 𝑅)) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}) → (^◡𝐹 “ 𝑧) ∈ 𝑅))
146	145	rexlimdvva 3038	. . . 4 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (∃𝑘 ∈ 𝒫 ∪ 𝑆∃𝑣 ∈ (𝑆 ×_t 𝑅)((𝑆 ↾_t 𝑘) ∈ Comp ∧ 𝑧 = {𝑓 ∈ (𝑆 Cn (𝑆 ×_t 𝑅)) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}) → (^◡𝐹 “ 𝑧) ∈ 𝑅))
147	13, 146	syl5bi 232	. . 3 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑧 ∈ ran (𝑘 ∈ {𝑤 ∈ 𝒫 ∪ 𝑆 ∣ (𝑆 ↾_t 𝑤) ∈ Comp}, 𝑣 ∈ (𝑆 ×_t 𝑅) ↦ {𝑓 ∈ (𝑆 Cn (𝑆 ×_t 𝑅)) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}) → (^◡𝐹 “ 𝑧) ∈ 𝑅))
148	147	ralrimiv 2965	. 2 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → ∀𝑧 ∈ ran (𝑘 ∈ {𝑤 ∈ 𝒫 ∪ 𝑆 ∣ (𝑆 ↾_t 𝑤) ∈ Comp}, 𝑣 ∈ (𝑆 ×_t 𝑅) ↦ {𝑓 ∈ (𝑆 Cn (𝑆 ×_t 𝑅)) ∣ (𝑓 “ 𝑘) ⊆ 𝑣})(^◡𝐹 “ 𝑧) ∈ 𝑅)
149		simpl 473	. . 3 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → 𝑅 ∈ (TopOn‘𝑋))
150		ovex 6678	. . . . . 6 ⊢ (𝑆 Cn (𝑆 ×_t 𝑅)) ∈ V
151	150	pwex 4848	. . . . 5 ⊢ 𝒫 (𝑆 Cn (𝑆 ×_t 𝑅)) ∈ V
152	9, 10, 11	xkotf 21388	. . . . . 6 ⊢ (𝑘 ∈ {𝑤 ∈ 𝒫 ∪ 𝑆 ∣ (𝑆 ↾_t 𝑤) ∈ Comp}, 𝑣 ∈ (𝑆 ×_t 𝑅) ↦ {𝑓 ∈ (𝑆 Cn (𝑆 ×_t 𝑅)) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}):({𝑤 ∈ 𝒫 ∪ 𝑆 ∣ (𝑆 ↾_t 𝑤) ∈ Comp} × (𝑆 ×_t 𝑅))⟶𝒫 (𝑆 Cn (𝑆 ×_t 𝑅))
153		frn 6053	. . . . . 6 ⊢ ((𝑘 ∈ {𝑤 ∈ 𝒫 ∪ 𝑆 ∣ (𝑆 ↾_t 𝑤) ∈ Comp}, 𝑣 ∈ (𝑆 ×_t 𝑅) ↦ {𝑓 ∈ (𝑆 Cn (𝑆 ×_t 𝑅)) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}):({𝑤 ∈ 𝒫 ∪ 𝑆 ∣ (𝑆 ↾_t 𝑤) ∈ Comp} × (𝑆 ×_t 𝑅))⟶𝒫 (𝑆 Cn (𝑆 ×_t 𝑅)) → ran (𝑘 ∈ {𝑤 ∈ 𝒫 ∪ 𝑆 ∣ (𝑆 ↾_t 𝑤) ∈ Comp}, 𝑣 ∈ (𝑆 ×_t 𝑅) ↦ {𝑓 ∈ (𝑆 Cn (𝑆 ×_t 𝑅)) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}) ⊆ 𝒫 (𝑆 Cn (𝑆 ×_t 𝑅)))
154	152, 153	ax-mp 5	. . . . 5 ⊢ ran (𝑘 ∈ {𝑤 ∈ 𝒫 ∪ 𝑆 ∣ (𝑆 ↾_t 𝑤) ∈ Comp}, 𝑣 ∈ (𝑆 ×_t 𝑅) ↦ {𝑓 ∈ (𝑆 Cn (𝑆 ×_t 𝑅)) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}) ⊆ 𝒫 (𝑆 Cn (𝑆 ×_t 𝑅))
155	151, 154	ssexi 4803	. . . 4 ⊢ ran (𝑘 ∈ {𝑤 ∈ 𝒫 ∪ 𝑆 ∣ (𝑆 ↾_t 𝑤) ∈ Comp}, 𝑣 ∈ (𝑆 ×_t 𝑅) ↦ {𝑓 ∈ (𝑆 Cn (𝑆 ×_t 𝑅)) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}) ∈ V
156	155	a1i 11	. . 3 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → ran (𝑘 ∈ {𝑤 ∈ 𝒫 ∪ 𝑆 ∣ (𝑆 ↾_t 𝑤) ∈ Comp}, 𝑣 ∈ (𝑆 ×_t 𝑅) ↦ {𝑓 ∈ (𝑆 Cn (𝑆 ×_t 𝑅)) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}) ∈ V)
157	9, 10, 11	xkoval 21390	. . . 4 ⊢ ((𝑆 ∈ Top ∧ (𝑆 ×_t 𝑅) ∈ Top) → ((𝑆 ×_t 𝑅) ^_ko 𝑆) = (topGen‘(fi‘ran (𝑘 ∈ {𝑤 ∈ 𝒫 ∪ 𝑆 ∣ (𝑆 ↾_t 𝑤) ∈ Comp}, 𝑣 ∈ (𝑆 ×_t 𝑅) ↦ {𝑓 ∈ (𝑆 Cn (𝑆 ×_t 𝑅)) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}))))
158	67, 70, 157	syl2anc 693	. . 3 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → ((𝑆 ×_t 𝑅) ^_ko 𝑆) = (topGen‘(fi‘ran (𝑘 ∈ {𝑤 ∈ 𝒫 ∪ 𝑆 ∣ (𝑆 ↾_t 𝑤) ∈ Comp}, 𝑣 ∈ (𝑆 ×_t 𝑅) ↦ {𝑓 ∈ (𝑆 Cn (𝑆 ×_t 𝑅)) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}))))
159		eqid 2622	. . . . 5 ⊢ ((𝑆 ×_t 𝑅) ^_ko 𝑆) = ((𝑆 ×_t 𝑅) ^_ko 𝑆)
160	159	xkotopon 21403	. . . 4 ⊢ ((𝑆 ∈ Top ∧ (𝑆 ×_t 𝑅) ∈ Top) → ((𝑆 ×_t 𝑅) ^_ko 𝑆) ∈ (TopOn‘(𝑆 Cn (𝑆 ×_t 𝑅))))
161	67, 70, 160	syl2anc 693	. . 3 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → ((𝑆 ×_t 𝑅) ^_ko 𝑆) ∈ (TopOn‘(𝑆 Cn (𝑆 ×_t 𝑅))))
162	149, 156, 158, 161	subbascn 21058	. 2 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝑅 Cn ((𝑆 ×_t 𝑅) ^_ko 𝑆)) ↔ (𝐹:𝑋⟶(𝑆 Cn (𝑆 ×_t 𝑅)) ∧ ∀𝑧 ∈ ran (𝑘 ∈ {𝑤 ∈ 𝒫 ∪ 𝑆 ∣ (𝑆 ↾_t 𝑤) ∈ Comp}, 𝑣 ∈ (𝑆 ×_t 𝑅) ↦ {𝑓 ∈ (𝑆 Cn (𝑆 ×_t 𝑅)) ∣ (𝑓 “ 𝑘) ⊆ 𝑣})(^◡𝐹 “ 𝑧) ∈ 𝑅)))
163	8, 148, 162	mpbir2and 957	1 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → 𝐹 ∈ (𝑅 Cn ((𝑆 ×_t 𝑅) ^_ko 𝑆)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ∃wrex 2913 {crab 2916 Vcvv 3200 ∩ cin 3573 ⊆ wss 3574 𝒫 cpw 4158 {csn 4177 ⟨cop 4183 ∪ cuni 4436 ∪ ciun 4520 ↦ cmpt 4729 × cxp 5112 ^◡ccnv 5113 dom cdm 5114 ran crn 5115 “ cima 5117 Fun wfun 5882 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ↦ cmpt2 6652 ficfi 8316 ↾_t crest 16081 topGenctg 16098 Topctop 20698 TopOnctopon 20715 Cn ccn 21028 Compccmp 21189 ×_t ctx 21363 ^_ko cxko 21364

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-fin 7959 df-fi 8317 df-rest 16083 df-topgen 16104 df-top 20699 df-topon 20716 df-bases 20750 df-cn 21031 df-cnp 21032 df-cmp 21190 df-tx 21365 df-xko 21366

This theorem is referenced by: cnmpt2k 21491

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